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Old 2003-09-06, 11:38   #1
gbvalor
 
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Aug 2002

3·37 Posts
Default Primes for a mersenne integer DWT FNT

This last week I left an Athlon XP searching primes in the form
[code:1]
p = k * 2^q + 1
[/code:1]
to be used in a FNT transform. For an integer DWT to be used in a modular Mersenne arithmetic we also need the existence of proper roots of two. I wrote a GMP based code to search systematicaly these primes. Is not too much optimized but is not slow ( about 800000 trials per second in my Athlon XP 2500+). You can download at

ftp://ftp.oxixares.com/pub/glucas/ilucas-0.1.tar.gz

After configure and make, you can read a brief help

./psearch -h

I've only tested psearch in X86 targets.

I made a long run searching all primes having at least 2^24 th root of two for numbers from 25 to 62 bits. The list is long. What follows is the better ones (with roots of two and unity at least of 2^26). The format is, for a s*2^r valid root:
p (hexadecimal), p(decimal), s, r, small root of unity of order p-1, a root of unity of order s*2^r, a root of two of order s*2^r, and the factorization of p-1.

[code:1]
0x000ff04e78000001, 4486344462041089, 1, 26, 11, 2956743534207098, 2534241762335360, (7193*1549*3*2^27)
0x01dda25a30000001, 134442072127045633, 1, 28, 5, 14202908006974440, 38416365583369715, (5323*397*79*3*2^28)
0x02dbd19908000001, 205988663169908737, 1, 26, 5, 205669437681594801, 134613325924526488, (41467*73*13^2*3*2^27)
0x040e0cc4c8000001, 292185065134227457, 1, 26, 19, 158881768599504245, 44348138445373279, (241883153*3^2*2^27)
0x07e6d93540000001, 569381225639182337, 1, 28, 3, 146665363182891925, 373737386647402321, (530277589*2^30)
0x07fffaece4000001, 576455172671143937, 1, 26, 3, 339740298459922762, 106826103507306054, (300439*28591*2^26)
0x089f279ef8000001, 621258837319483393, 3, 26, 23, 247430348120354218, 461368420807572000, (3129641*29*17*3*2^27)
0x099649b6e8000001, 690820642773139457, 1, 27, 3, 334361055820297156, 646845165018050406, (302765581*17*2^27)
0x09f6e9ff34000001, 718018472910520321, 49, 26, 31, 327752487228664797, 464703741142552395, (8429*157*11*7^2*5*3*2^26)
0x0ac9ad8038000001, 777343176905064449, 1, 27, 3, 630431347559292957, 208289648825889548, (4409*199*41*23*7*2^27)
0x0b1052f75c000001, 797228356398481409, 1, 26, 3, 392632981377933164, 483000991053494976, (1422199*8353*2^26)
0x0c201b53e4000001, 873728374831316993, 1, 26, 3, 361688085632418079, 645118062240714450, (34429*1531*19*13*2^26)
0x0cd5994428000001, 924813816502419457, 3, 26, 7, 554684369476632643, 238614840738348519, (2296799959*3*2^27)
0x0d8df8f49c000001, 976710423682220033, 1, 26, 5, 278474532125113655, 111509898764842557, (5682983*197*13*2^26)
0x0dc75649c4000001, 992857117679419393, 1, 26, 13, 113145346672272579, 503552499055848415, (182650913*3^4*2^26)
0x0de7ac3278000001, 1001958774874832897, 1, 26, 3, 746619134835021329, 157588362536481380, (198031*149*23*11*2^27)
0x0e3622a4a4000001, 1024044055794810881, 1, 26, 3, 488269427790767265, 152470912434199161, (50030969*61*5*2^26)
0x0f3c18d6a4000001, 1097779719325089793, 1, 26, 19, 264312382562420289, 438304236734466310, (2228333*2447*3*2^26)
0x111ddc590c000001, 1233384147980910593, 1, 26, 3, 377843144979909225, 447045291288505038, (62726471*293*2^26)
0x12a5b7d1c8000001, 1343682175471452161, 1, 27, 3, 334429903422378648, 81492114934364874, (2002242469*5*2^27)
0x13669ae0cc000001, 1397975024631021569, 1, 26, 3, 1157281415442149401, 788870162067539560, (31231561*29*23*2^26)
0x14acf50dfc000001, 1489835017164029953, 3, 26, 17, 576403982379777951, 1217496306594615948, (569237737*13*3*2^26)
0x14c7a26f1c000001, 1497343999218810881, 1, 26, 3, 826087831612280885, 3699770312001190, (40939751*109*5*2^26)
0x1533ee1c5c000001, 1527826504178728961, 5, 26, 3, 961536035178335096, 918872096539104929, (4553277803*5*2^26)
0x15508b7d38000001, 1535880842860036097, 1, 26, 3, 1037146288658234910, 981618541284570884, (2437*1873*109*23*2^27)
0x16110c4a58000001, 1590065656882003969, 1, 27, 7, 795498821589657901, 563849041443623430, (3948970777*3*2^27)
0x169e44eccc000001, 1629815898980220929, 1, 26, 3, 1293620391333236633, 738488026520989437, (307419613*79*2^26)
0x173355a7d4000001, 1671774065989255169, 1, 26, 3, 253500402584444749, 125899044871034282, (673280401*37*2^26)
0x17eafac418000001, 1723465527545823233, 1, 27, 3, 1216173920984816684, 975283400163867802, (15602453*823*2^27)
0x185ec3c8f0000001, 1756056172508479489, 3, 27, 11, 1066150737387558173, 328123169860203926, (13057523*167*3*2^28)
0x18e546661c000001, 1793917430947381249, 9, 26, 19, 1199447079096636291, 280060442015752002, (2970161423*3^2*2^26)
0x196c4bc084000001, 1831922438653345793, 1, 26, 3, 491432681299295688, 222744005175096612, (4733*911*487*13*2^26)
0x1978a65fc8000001, 1835399778461024257, 1, 26, 7, 543492426188099148, 1652527452479821756, (12986509*13*3^4*2^27)
0x1bc631e684000001, 2001341950539595777, 7, 26, 43, 1892824508735447530, 135416102482748349, (156797*3019*7*3^2*2^26)
0x1cb3ec2228000001, 2068256285337321473, 1, 27, 3, 381679717147806527, 592009946307101677, (375846589*41*2^27)
0x1f3c2e2bb4000001, 2250724679016775681, 1, 26, 23, 1120209764861288188, 211656614197246419, (321203*6961*5*3*2^26)
0x1f832d661c000001, 2270708553704538113, 1, 26, 5, 1362403533105130967, 2001390660479887798, (3564331*863*11*2^26)
0x21499e999c000001, 2398622659135930369, 9, 26, 23, 861431578568445555, 576311146788870038, (8964701*443*3^2*2^26)
0x227e1d06e4000001, 2485455959787307009, 3, 26, 7, 681888733638616278, 1522226130946868853, (12345393299*3*2^26)
0x23881e62d4000001, 2560329797972525057, 1, 26, 3, 1421751396220275612, 2426702380623961098, (38151887029*2^26)
0x23c33e1230000001, 2576971659621695489, 1, 26, 3, 47991005390020579, 2567531320003683670, (45013*509*419*2^28)
0x23d54cecc4000001, 2582054541151174657, 1, 26, 5, 335167499971725653, 105217944468263621, (472331*431*7*3^3*2^26)
0x24e3241724000001, 2658007886885552129, 1, 26, 3, 17087961293468997, 2502952362987393078, (747309509*53*2^26)
0x250655a3ac000001, 2667913990717308929, 7, 26, 3, 1408772344356920451, 1708937673388724268, (228203*607*41*7*2^26)
0x2543e417f0000001, 2685240594304860161, 1, 28, 7, 437521231975298786, 2246643140016868175, (418811*281*17*5*2^28)
0x263381a61c000001, 2752686347687821313, 1, 26, 3, 1211497281217223080, 582120517783096743, (1000444463*41*2^26)
0x270b8fcc50000001, 2813500499901284353, 1, 26, 5, 1493683036924742258, 1803913865741770605, (2349497*1487*3*2^28)
0x27d65c0dcc000001, 2870583026822152193, 1, 26, 3, 1544040082115877118, 1267223265068524320, (1799311*23773*2^26)
0x28c6855830000001, 2938182420714749953, 1, 26, 5, 1443447788221454795, 2203004921239146984, (3648527489*3*2^28)
0x29372c6c20000001, 2969891322179092481, 13, 26, 3, 246015212717434719, 78065254858085547, (85105441*13*5*2^29)
0x2976921ce4000001, 2987736055588847617, 1, 26, 5, 814499116290532621, 1395867229455529121, (1289447*677*17*3*2^26)
0x299da4c304000001, 2998734084391305217, 1, 26, 7, 2193404549758534209, 2280611878894958909, (79651729*17*11*3*2^26)
0x29e0138ec4000001, 3017433254232850433, 1, 26, 3, 493867084350267467, 1968449104587829253, (917617537*7^2*2^26)
0x2b0985cd0c000001, 3101156934137413633, 1, 26, 7, 2090635873337008584, 521344467958667567, (31793*101*41*13*3^3*2^26)
0x2b8a61fd88000001, 3137427831953293313, 1, 26, 5, 1536091008234844030, 1238193593117611775, (3339379847*7*2^27)
0x2c16298620000001, 3176772243195691009, 9, 27, 23, 2998300564021550191, 902574819759800553, (979831*61*11*3^2*2^29)
0x2c842901d0000001, 3207733922356002817, 7, 28, 5, 1523683635041096692, 3154971583035042370, (297457*1913*7*3*2^28)
0x2ca26792d4000001, 3216246964215611393, 1, 26, 3, 1517518663274324905, 2129119363349800761, (92951*2467*19*11*2^26)
0x2d1c07a768000001, 3250481446640418817, 1, 27, 5, 1065810059449338785, 1623720257175915718, (6529*1021*173*7*3*2^27)
0x2db442da08000001, 3293330731719196673, 7, 27, 3, 113330352708327626, 2298372395216713739, (9521*1061*347*7*2^27)
0x2eeb9eb840000001, 3380970459456602113, 1, 26, 5, 997561900314377058, 2059917662335632305, (1536737*683*3*2^30)
0x311e372e88000001, 3539327030149513217, 1, 26, 3, 1203980672071061596, 105697814525703480, (497547949*53*2^27)
0x31cd9da590000001, 3588697812495237121, 1, 27, 17, 1209348937195381431, 1570138288680028516, (633449*67*7*5*3^2*2^28)
0x31e8242734000001, 3596164053250080769, 3, 26, 11, 1565827433382961282, 1196303003056152634, (1050725887*17*3*2^26)
0x328be81b78000001, 3642259928334204929, 1, 27, 3, 1443726139595968925, 3278532423853235697, (9024593*97*31*2^27)
0x335ffde83c000001, 3701956592602710017, 1, 26, 3, 2186188583595948928, 1320013042145138993, (2398410953*23*2^26)
0x33daf1d008000001, 3736564716646957057, 1, 27, 13, 1650854658436879311, 2393133708681984307, (10581367*877*3*2^27)
0x351bda0968000001, 3826892042315759617, 3, 26, 5, 2185156650844579829, 2538502083054451593, (1059671*8969*3*2^27)
0x3730f281e4000001, 3976945110658056193, 1, 26, 5, 2993732503061540936, 1039955707259381352, (19753700051*3*2^26)
0x37bcd30b34000001, 4016317012778156033, 1, 26, 3, 465041681940351762, 507119766593844459, (27197*17327*127*2^26)
0x37c62feb48000001, 4018952405046525953, 1, 27, 3, 1802636177853445231, 3729246111913317172, (14845577*2017*2^27)
0x38b70d964c000001, 4086750126033338369, 1, 26, 3, 1236222603584416280, 3528491310214083276, (1409*1229*139*23*11*2^26)
0x3995cfeabc000001, 4149451238775193601, 1, 26, 3, 2773114793432805883, 2137113916727713022, (70853*521*67*5^2*2^26)
0x3a25874234000001, 4189903746748121089, 1, 26, 41, 1997758893125550556, 92971526868374832, (223779323*31*3^2*2^26)
0x3b84703054000001, 4288676098033582081, 1, 26, 13, 4087210687920489658, 3864104511162925379, (608630989*7*5*3*2^26)
0x3b9e52c790000001, 4295962111625920513, 1, 28, 5, 1278672322965972673, 1588269684189040326, (386647*73*7*3^4*2^28)
0x3c8d9159fc000001, 4363303429657853953, 1, 26, 7, 830487614483445701, 3435199924203284627, (3433037*107*59*3*2^26)
0x3d4853f504000001, 4415871746435973121, 5, 26, 37, 3165491059413527341, 2004072930527956147, (92507*479*11*5*3^3*2^26)
0x3d68485748000001, 4424866173598826497, 3, 27, 5, 3222940107848139080, 1956734948064434138, (10989274019*3*2^27)
0x3d8c0af1c8000001, 4434931766611935233, 1, 27, 3, 2481622260952398909, 3091977614176773732, (1135531*4157*7*2^27)
0x3e08fdd1a8000001, 4470101707073781761, 5, 26, 3, 914378621753178986, 3145105531751226428, (269141*24749*5*2^27)
0x3e2eb5c6b0000001, 4480718544241164289, 7, 27, 19, 768648899009962081, 1836497973012950493, (10301*89*17^2*7*3^2*2^28)
0x3e9dd6954c000001, 4511998338408644609, 7, 26, 3, 1737434714250511834, 3611972301602738930, (9604859221*7*2^26)
0x3ef166f03c000001, 4535519481723617281, 1, 26, 7, 2981000636658545702, 3851560279867545994, (2273*857*257*5*3^3*2^26)
0x3ef71afcf8000001, 4537124823393894401, 1, 26, 3, 3624280287339433243, 1763783591557992766, (1979749*683*5^2*2^27)
0x3f20ebbbb0000001, 4548894814988402689, 1, 27, 7, 4281543534721332498, 3107376785910416954, (63463*29669*3^2*2^28)
0x3fae7cee80000001, 4588742434128658433, 1, 29, 3, 132610359250039243, 3762940913832781428, (194254519*11*2^31)
[/code:1]

Guillermo
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Old 2003-09-08, 16:05   #2
gbvalor
 
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Aug 2002

3·37 Posts
Default

And these are the primes p
[code:1]2^62 < p < 2^63[/code:1]
with at least root of two and unity of order k*2^26. Enjoy ;)
[code:1]
0x41be225448000001, 4737261603420897281, 5, 27, 7, 1178821920194683392, 4675487877597713469, (1069*1051*103*61*5*2^27)
0x41defd8784000001, 4746509815772086273, 1, 26, 5, 3089518248187704353, 3918102132413537936, (12119*1667*389*3^2*2^26)
0x42d2d93564000001, 4815149774946697217, 1, 26, 3, 3818589844401922, 1361689833336711323, (183611*181*127*17*2^26)
0x4381eb910c000001, 4864428080740958209, 1, 26, 11, 2958967816062370172, 2655787245967634374, (114511259*211*3*2^26)
0x445d704e28000001, 4926217048394039297, 1, 27, 3, 1027948736703355701, 2131531404765743653, (6563*523*37*17^2*2^27)
0x44af5f5ea8000001, 4949279375653601281, 1, 27, 7, 584789657756515863, 1732921441466631660, (819444553*5*3^2*2^27)
0x455f926bd0000001, 4998875103153356801, 1, 28, 3, 1980171811511315460, 1811601276916892925, (120163*6199*5^2*2^28)
0x45d0fe37b0000001, 5030800298902290433, 1, 26, 5, 3253410958134744929, 2317319277548422372, (799573*601*13*3*2^28)
0x45d41c7f14000001, 5031678015801655297, 1, 26, 5, 3665593605573574446, 3858338049521078410, (1255721*1531*13*3*2^26)
0x470fd0b850000001, 5120540791378280449, 1, 26, 7, 578423857485044221, 2493707208463744283, (6358500311*3*2^28)
0x47924b9b54000001, 5157267653793939457, 1, 26, 5, 787762103660387305, 1117991535234788083, (25616425543*3*2^26)
0x4799d0c7c8000001, 5159384404602126337, 1, 26, 13, 4156205608933942035, 4622381058766144496, (998167*389*11*3^2*2^27)
0x47ee2e5148000001, 5183131147785207809, 1, 27, 3, 1796452599409217936, 1552294913820480033, (38617336361*2^27)
0x4814e10f04000001, 5194023724872892417, 1, 26, 5, 492821443618634129, 93899412453812596, (8609*3779*61*13*3*2^26)
0x487e32cfbc000001, 5223668485590089729, 1, 26, 3, 2991478112381473433, 3675272241482776912, (59555261*1307*2^26)
0x48bfdac4c4000001, 5242149029919326209, 1, 26, 7, 4679899253435527391, 3073118625132507951, (1608577*16187*3*2^26)
0x4b7e657ffc000001, 5439897000273248257, 3, 26, 5, 2460097516872013219, 5238729617274691959, (27020260693*3*2^26)
0x4c2f87f388000001, 5489755950816428033, 7, 26, 3, 720003830589051749, 5435130787642042252, (29362433*199*7*2^27)
0x4c6b0eda8c000001, 5506511301205032961, 1, 26, 11, 4153023033857221012, 409900373162118585, (1823409167*5*3^2*2^26)
0x4c7e73e318000001, 5511970413145489409, 1, 26, 3, 245393816109111676, 5435420923031351478, (472457*86923*2^27)
0x4da01408f8000001, 5593492765947199489, 3, 26, 11, 4235931373525975985, 4238096017709711744, (83183171*167*3*2^27)
0x4e232ac268000001, 5630390973600104449, 1, 26, 17, 3979588141028309871, 3987866330721086265, (25577*16567*11*3^2*2^27)
0x4e5a7a4f9c000001, 5645959565200588801, 25, 26, 7, 598947704866249836, 1742787163194063069, (609979*613*5^2*3^2*2^26)
0x4e90e1607c000001, 5661272536118198273, 1, 26, 3, 2697535678017370767, 5121844923198538990, (927027853*13*7*2^26)
0x4f98cd6848000001, 5735559973224579073, 1, 27, 5, 2218879627766196117, 5684558643507354333, (33911*24709*17*3*2^27)
0x4fb17733a0000001, 5742502063485485057, 1, 27, 3, 199663621977240601, 5566904116653695975, (83117*11699*11*2^29)
0x5197d2da84000001, 5879399674514767873, 1, 26, 11, 3935451098611529720, 775981491142654490, (29203294091*3*2^26)
0x51d01dbfe8000001, 5895244622296252417, 1, 26, 5, 987729580102736825, 2653754078487539817, (62836903*233*3*2^27)
0x5356900354000001, 6005145497113722881, 1, 26, 3, 2402826101567678698, 2457124396017407595, (1317971*367*37*5*2^26)
0x54add68484000001, 6101768934795509761, 3, 26, 13, 2970261022315879639, 5491437271761955712, (6061562831*5*3*2^26)
0x556b213210000001, 6155049814632431617, 3, 26, 5, 197268383298387548, 3485828664008760265, (130253*58679*3*2^28)
0x55c0ff5a2c000001, 6179219451502657537, 1, 26, 5, 397151336475474759, 5667341485858749483, (99975619*307*3*2^26)
0x55c8f31f54000001, 6181457804444827649, 1, 26, 3, 3509426506206497932, 5992098597932704925, (2489483761*37*2^26)
0x55d6de55d8000001, 6185375598517420033, 1, 26, 17, 4811348399177529839, 3933272000326523551, (18756467*13*7*3^3*2^27)
0x568bc751fc000001, 6236297263958720513, 1, 26, 3, 3971943100011612382, 2969568566470012082, (513923*2477*73*2^26)
0x56ba086870000001, 6249316677578653697, 1, 28, 3, 16232144420994102, 910120255520714019, (23280518791*2^28)
0x56c3909a04000001, 6251999698886852609, 1, 26, 3, 4043724965909581224, 4641972566691666386, (12143*9277*827*2^26)
0x57a1b7f9a0000001, 6314530435309240321, 1, 27, 29, 3638151391989654192, 5714914180085568448, (27073*2633*11*5*3*2^29)
0x5811262e60000001, 6345895330562703361, 1, 26, 17, 5293835368098253656, 1173405048301900746, (13356103*59*5*3*2^29)
0x5825e4f7ac000001, 6351734601869688833, 1, 26, 3, 431928824266055419, 5430710149583430048, (8039431*193*61*2^26)
0x5828a131dc000001, 6352504509922541569, 1, 26, 31, 3186506404906029160, 3758171405228665233, (269685737*13*3^3*2^26)
0x589d63590c000001, 6385369080785797121, 1, 26, 3, 3762241922311240736, 3702680255630927189, (31350713*607*5*2^26)
0x58aab3ad8c000001, 6389116579337994241, 5, 26, 7, 1291311601295705051, 5051972391164229564, (6404659*991*5*3*2^26)
0x5a45dcb3e8000001, 6504847902056120321, 1, 27, 3, 19761297724459081, 161288939930402354, (133979*6577*11*5*2^27)
0x5d25804e00000001, 6711911892161396737, 1, 27, 17, 1725083916306068501, 6410363315800515134, (17159*353*43*3*2^33)
0x5d72e9974c000001, 6733701228896714753, 1, 26, 3, 1450660986791701175, 2155015198645221293, (942593*106451*2^26)
0x5e573ff5e4000001, 6797972487863795713, 1, 26, 5, 926593750423538974, 2895595734026364624, (3583*3343*2819*3*2^26)
0x5f3d18af44000001, 6862668548221698049, 1, 26, 11, 3403421891251666208, 5477493769838844568, (1262490497*3^4*2^26)
0x5f71c6d6e8000001, 6877496732289073153, 1, 27, 11, 1405974239287623767, 2724485284034045627, (28723*10079*59*3*2^27)
0x601b82ae08000001, 6925272535982407681, 5, 27, 17, 5982985666948329794, 4545843944735538143, (5235647*73*5*3^3*2^27)
0x619445239c000001, 7031320937475604481, 1, 26, 3, 2625323403112415849, 2884709133791257056, (7096163*2953*5*2^26)
0x61b4ff59cc000001, 7040532878626652161, 1, 26, 47, 3895507947877450550, 560890805191168742, (9892703*101*7*5*3*2^26)
0x61d595ac20000001, 7049705358222163969, 1, 29, 7, 6063674285037906815, 6520630918353253221, (1220249*211*17*3*2^29)
0x6217c85154000001, 7068338391808475137, 1, 26, 7, 6144100978384966088, 3077919810079936224, (18200527*643*3^2*2^26)
0x62ec903f20000001, 7128230911014862849, 1, 29, 11, 4540683410978582655, 4766754867344035959, (25582591*173*3*2^29)
0x634ccba2d0000001, 7155317808119676929, 1, 26, 3, 2947552095399469628, 6193470200279678203, (562201*2789*17*2^28)
0x63efaa7c08000001, 7201161778875596801, 1, 27, 3, 7171160407115174605, 1082331540358995760, (3535607*607*5^2*2^27)
0x64351c6074000001, 7220708778147053569, 1, 26, 11, 719610484328719715, 6272875778789361851, (749653*47843*3*2^26)
0x64522e7cb4000001, 7228891465013788673, 1, 26, 3, 2460199783617487631, 3127807033921820210, (65562311*53*31*2^26)
0x64e622ec78000001, 7270537047457136641, 5, 27, 7, 5434380278030048302, 1840007641395133088, (20071*1487*11^2*5*3*2^27)
0x64f4f7d944000001, 7274711810631008257, 3, 26, 5, 5191496054571508870, 7723892966058422, (302191*1097*109*3*2^26)
0x654222072c000001, 7296431760491937793, 3, 26, 13, 4588621106149969955, 3924224747762560468, (38270083*947*3*2^26)
0x65431c3150000001, 7296706819391488001, 1, 28, 7, 2733716872191064998, 2999048657769929536, (289559*751*5^3*2^28)
0x6552beb420000001, 7301107626761256961, 1, 29, 37, 5421046896538769723, 1711743994037358230, (906624847*5*3*2^29)
0x6655eb4460000001, 7374058643789971457, 1, 26, 3, 1396027201665501323, 3078416317400299104, (31649*18869*23*2^29)
0x66ee077688000001, 7416873841995022337, 1, 26, 3, 1557909721899588172, 188779960122487356, (1285116659*43*2^27)
0x67c7f8d80c000001, 7478219313069948929, 1, 26, 3, 4795173481076462275, 806968439478416296, (5393*1979*197*53*2^26)
0x68ec944e04000001, 7560580937243688961, 3, 26, 7, 537205835435945602, 6463750973640390217, (3697*1933*1051*5*3*2^26)
0x69957e49fc000001, 7608126001737891841, 15, 26, 19, 4375800430050353091, 1259565773241687670, (47147*22901*7*5*3*2^26)
0x69a4286e20000001, 7612253723611365377, 1, 29, 3, 6377111507559252754, 5694969317044767112, (11945179*1187*2^29)
0x6a63cdfe48000001, 7666197482728062977, 1, 26, 3, 83342443016876116, 4597554753367588153, (3187901*41*23*19*2^27)
0x6c303d5e64000001, 7795798430592270337, 3, 26, 5, 1937068944536284377, 7085157817978947170, (86951*563*113*7*3*2^26)
0x6e13065920000001, 7931690348588302337, 1, 29, 3, 4205310525794250888, 811224012355076735, (1579*811*139*83*2^29)
0x6e6fcc79cc000001, 7957803890072616961, 1, 26, 7, 3393710621489957002, 104694678383806615, (73881943*107*5*3*2^26)
0x6ed2869c14000001, 7985593094189875201, 1, 26, 29, 2831696102990925043, 2160073427282412235, (1487*1151*103*5^2*3^3*2^26)
0x6f3b22492c000001, 8015037659502084097, 3, 26, 7, 1734656278755605982, 4422318424766964088, (525719*1847*41*3*2^26)
0x6f4570fc9c000001, 8017938941854351361, 1, 26, 3, 1726857618026335356, 4020881010881241994, (1316039*271*67*5*2^26)
0x717a3115f8000001, 8176902043869380609, 1, 26, 3, 7889722325825468484, 4029839328403873030, (303731*2539*79*2^27)
0x719630ba74000001, 8184782950160662529, 1, 26, 3, 4136912489775897206, 6620854485775752156, (578022607*211*2^26)
0x721d957814000001, 8222892837612552193, 3, 26, 5, 6182349702240995749, 1526232670771237224, (517006969*79*3*2^26)
0x72ab3705f0000001, 8262758439982202881, 3, 28, 37, 6113744652447966378, 5697423558884340765, (2052078257*5*3*2^28)
0x72d5ef868c000001, 8274783251516620801, 1, 26, 23, 1244728577971414608, 5539022303919125863, (126465517*13*5^2*3*2^26)
0x747e5c0b0c000001, 8394247957980905473, 1, 26, 5, 6572213061088923433, 1035096925252868240, (22279*389*283*17*3*2^26)
0x75054963a8000001, 8432226569690284033, 1, 27, 23, 5806145828944417800, 927221919742760714, (1990463*167*7*3^3*2^27)
0x751dba6cd4000001, 8439106253338181633, 1, 26, 3, 7186714441757300164, 7721556239409027174, (125752482613*2^26)
0x758482f788000001, 8468037199013609473, 3, 27, 5, 6151008754265944653, 6967300399021635203, (930187*983*23*3*2^27)
0x7646a40860000001, 8522679700722548737, 1, 26, 5, 7470826507489444662, 6999187398166240446, (13673323*43*3^3*2^29)
0x7673664378000001, 8535278158759329793, 1, 27, 5, 3391745149082591718, 5589439648110667966, (290377981*73*3*2^27)
0x7724735920000001, 8585113616301162497, 1, 29, 3, 2156431095123872823, 739543445527210547, (9313349*101*17*2^29)
0x78168d401c000001, 8653259040526172161, 45, 26, 13, 3241651979860258156, 4441065042563055809, (260492137*11*5*3^2*2^26)
0x79200bd3fc000001, 8727989082937884673, 3, 26, 37, 8559321344775507776, 8114760517139229000, (358284221*11^2*3*2^26)
0x7930ddbb90000001, 8732723475115933697, 13, 28, 3, 1036906468684498247, 2101233126731816791, (2293727*1091*13*2^28)
0x7931fa9964000001, 8733036689162764289, 1, 26, 3, 586508017026383012, 5142243070691083499, (6733*1987*137*71*2^26)
0x7a4b404564000001, 8812207762655150081, 5, 26, 3, 5747184562960155142, 3463874976030873359, (54827611*479*5*2^26)
0x7aea816c2c000001, 8857033918772150273, 1, 26, 3, 6215936195698917148, 1908822655421748514, (3842327*701*7^2*2^26)
0x7b1c168a10000001, 8870990148242767873, 1, 26, 7, 8173957334927840857, 8223274471822110163, (3671890393*3^2*2^28)
0x7b500d9774000001, 8885617008939368449, 1, 26, 7, 2923430012765137436, 4514297480817895666, (159333347*277*3*2^26)
0x7c03e0d770000001, 8936233301534834689, 1, 26, 29, 4201458454597494322, 2056974331796584691, (1767833*6277*3*2^28)
0x7c2799e55c000001, 8946288395164844033, 1, 26, 3, 6927979828421191833, 937512584777075049, (325187*3761*109*2^26)
0x7ccdc48b74000001, 8993060134154403841, 5, 26, 53, 8548169858889952792, 755825226253428851, (911*139*67*13*5*3^5*2^26)
0x7d5d841654000001, 9033521659008516097, 3, 26, 23, 7604583044063181894, 382361391177247495, (179849*1229*29*7*3*2^26)
0x7d7b89641c000001, 9041971739935899649, 3, 26, 7, 2827030106195821977, 1413366923571613384, (169709*20357*13*3*2^26)
0x7de72d3530000001, 9072269680822976513, 1, 26, 3, 2143381757311453109, 2107600745602150436, (334493*191*23^2*2^28)
0x7ead9826b8000001, 9128119311813050369, 1, 26, 3, 1532048130115685181, 6281114441826074702, (3683971*18461*2^27)
0x7f536aecb4000001, 9174794430748557313, 13, 26, 5, 8692105388473398617, 192361831445389925, (2151943*181*13*3^3*2^26)
0x7f771d9034000001, 9184842470230327297, 1, 26, 5, 5273248505984149652, 5036985230344747716, (393083*3517*11*3^2*2^26)
0x7f7de52250000001, 9186750750439505921, 5, 26, 3, 5505911987368794467, 1300738607518816434, (60013*6709*17*5*2^28)
0x7f9475e978000001, 9193102384971120641, 5, 27, 19, 2540904207936979916, 532730828978699718, (204459553*67*5*2^27)
[/code:1]

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